Please answer all $5$ questions ::::::: Problem $7-20($Algo$)$

L$.$ Winston Martin $($an allergist$)$ has an excellent system for handling his regular patients who come in just for allergy injections. Patients arrive for an injection and fill out a name slip, which is then placed in an open slot that passes into another room staffed by up to three nurses. The specific injections for a patient are prepared, and the patient is called through a speaker system into the room to receive the injection. At certain times during the day, patient load drops and only one nurse is needed to administer the injections.

Let$$s focus on the simpler case of the two$$namely$,$ when there is one nurse. Also, assume that patients arrive in a Poisson fashion and the service rate of the nurse is exponentially distributed. During this slower period, patients arrive with an interarrival time of approximately $3$ minutes. It takes the nurse an average of $1\mathrm{.}80$ minutes to prepare the patients$$ serum and administer the injection. $($Use the Excel spreadsheet Queue Models.$)$

$1.$What is the average number of patients you would expect to see in Dr$.$ Martin's facilities?

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

$2.$How long would it take for a patient to arrive, get an injection, and leave?

Note: Round your answer to $2$ decimal places.

$3.$What is the probability that there will be three or more patients on the premises?

Note: Do not round intermediate calculations. Round your answer to $1$ decimal place.

$4.$What is the utilization of the nurse?

Note: Do not round intermediate calculations. Round your answer to $1$ decimal place.

$5.$ Assume three nurses are available. Each takes an average of $1\mathrm{.}80$ minutes to prepare the patients' serum and administer the injection. What is the average total time of a patient in the system?

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.